An extension theorem for strongly locally homogeneous continua

Let X be a strongly locally homogeneous continuum other than S¹. We prove that every homeomorphism between two countable compacta lying in X can be extended to a homeomorphism of the whole continuum X.     

A continuum with no prime shape factors

As the main result of this paper, we prove that there exists a continuum with non-trivial shape without any prime factor. This answers a question of K. Borsuk [K. Borsuk, Concerning the notion of the shape of compacta, in: Proc. Internat. Symposium on Topology and Its Applications, Herceg-Novi, 1968, pp. 98-104]. We also show that for each integer n?3 there exists a continuum X such that Sh(X,x)=Shn(X,x), but Sh(X,x)≠Shn−1(X,x). Therefore we obtain the negative answer to another question of K. Borsuk [K. Borsuk, Some remarks concerning the shape of pointed compacta, Fund. Math. 67 (1970) 221-240]: Does Sh(X,x)=Shn(X,x), for a compactum X and some integer n?3, implie that Sh(X,x)=Sh²(X,x)?     

On 2-dimensional Nonaspherical Cell-like Peano Continua: A Simplified Approach

We construct a functor AC(?, ?) from the category of path connected spaces X with a base point x to the category of simply connected spaces. The following are the main results of the paper: (i) If X is a Peano continuum then AC(X, x) is a cell-like Peano continuum; (ii) If X is n-dimensional then AC(X, x) is (n + 1)?dimensional; and (iii) For a path connected space X, π ₁(X, x) is trivial if and only if π ₂(AC(X, x)) is trivial. As a corollary, AC(S ¹, x) is a 2-dimensional nonaspherical cell-like Peano continuum.     

Embedding Tn-like continua in Euclidean space

Many authors have been concerned with embedding ∏-like continua in Rⁿ where ∏ is some collection of polyhedra or manifolds. A similar concern has been embedding ∏-like continua in Rⁿ up to shape. In this paper we prove two main theorems. Theorem: If n ? 2 and X is Tⁿ-like, then X embeds in R²ⁿ. This result was conjectured by McCord for the case H¹(X) finitely generated and proved by McCord for the case that H¹(X) = 0 using a theorem of Isbell. The second theorem is a shape embedding theorem. Theorem: If X is Tⁿ-like, then X embeds in Rⁿ⁺² up to shape. This theorem is proved by showing that an n-dimensional compact connected abelian topological group embeds in Rⁿ⁺². Any Tⁿ-like continuum is shape equivalent to a k-dimensional compact connected abelian topological group for some 0 ? k ? n.     

Products of Compacta with Plyhedra and Topological Spaces in the Shape Category

In the literature there exist examples of separable metric spaces X,Y whose Cartesian product X × Y is not a product in the shape category Sh(Top). It is an open question whether, for X a compact Hausdorff space, X × Y is a product in Sh(Top), for every topological spaces Y. The main result of the paper asserts that the answer is positive provided X × P is a product in Sh(Top), for every polyhedron P.     

Motivic equivalence of quadratic forms. II

Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X _φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X _φ and X _ψ coincide but . For a pair of anisotropic (2 ^{n -1})-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X. Received: 27 September 1999 / Revised version: 27 December 1999     

Certain finite dimensional maps and their application to hyperspaces

In [8] Y. Sternfeld and this author gave a positive answer to the following longstanding open problem: Is the hyperspace (=the space of all subcontinua endowed with the Hausdorff metric) of a 2-dimensional continuum infinite dimensional? This result was improved in [9] where it was shown that for every positive integer numbern a 2-dimensional continuum contains a 1-dimensional subcontiuum with hyperspace of dimension ≥n. And it was asked there: Does a 2-dimensional continuum contain a 1-dimensional subcontinuum with infinite dimensional hyperspace? In this note we answer this question in the positive. Our proof applies maps with the following properties. A real valued mapf on a compactumX is called a Bing map if every continuum that is contained in a fiber off is hereditarily indecomposable.f is called ann-dimensional Lelek map if the union of all non-trivial continua which are contained in the fibers off isn-dimensional. It is shown that for dimX=n+1 the maps which are both Bing andn-dimensional Lelek maps form a denseG _σ-subset of the function spaceC(X, I)     

Stabilization of 4-manifolds

The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×S2are not deformation equivalent.In this note,we show that the stabilized manifolds X×S1and Y×S1are diffeomorphic and non-deformation equivalent in cosymplectic sense.     

Quasi-monotone mappings on θn-continua

A continuous function f from a continuum X onto a continuum Y is quasi-monotone if, for every subcontinuum M of Y with nonvoid interior, f^-1(M) has a finite number of components each of which is mapped onto M by f. A θ_n-continuum is one that no subcontinuum separates into more than n components. It is known that if f is quasi-monotone and X is a θ₁-continuum, then Y is a θ₁-continuum or a θ₂-continuum that is irreducible between two points. Examples are given to show that this cannot be generalized to a θ_n-continuum and n + 1 points for any n >1, but it is proved that if f is quasi-monotone and X is a θ_n-continuum, then Y is a θ_n-continuum or a θ_n+1-continuum that is the union of n + 2 continua H,S₁,S₂,…,S_n+1, whe for each i, S_i is the closure of a component of Y H, S_i is irreducible from some point P_i to H, and H is irreducible about its boundary. Some theorems and examples are given concerning the preservation of decomposition elements by a quasi-monotone map defined on a θ_n-continuum that admits a monotone, upper-semicontinuous decomposition onto a finite graph.     

Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron

In a previous paper the author has associated with every inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution RK(X) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then RK(X) consists of spaces having the homotopy type of polyhedra. In the present paper it is proved that this construction is functorial. One of the consequences is the existence of a functor from the strong shape category of compact Hausdorff spaces X to the shape category of spaces, which maps X to the Cartesian product X×P. Another consequence is the theorem which asserts that, for compact Hausdorff spaces X, X^′, such that X is strong shape dominated by X^′ and the Cartesian product X^′×P is a direct product in Sh(Top), then also X×P is a direct product in the shape category Sh(Top).     

Testing for affine equivalence of elliptically symmetric distributions

Let X and Y be d-dimensional random vectors having elliptically symmetric distributions. Call X and Y affinely equivalent if Y has the same distribution as AX+b for some nonsingular d×d-matrix A and some . This paper studies a class of affine invariant tests for affine equivalence under certain moment restrictions. The test statistics are measures of discrepancy between the empirical distributions of the norm of suitably standardized data.     

Almost continuous orbit equivalence for non-singular homeomorphisms

Let X and Y be Polish spaces with non-atomic Borel measures μ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, μ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III ₁ or that they are both of type III _λ, 0 < λ < 1 and, in the III _λ case, suppose in addition that both ‘topological asymptotic ranges’ (defined in the article) are log λ · ℤ. Then there exist invariant dense G _δ-subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a non-singular homeomorphism ϕ: X′ → Y′ which is an orbit equivalence between T| _X′ and S| _Y′, that is ϕ{T ⁱ x} = {S ⁱ ϕx} for all x ∈ X′. Moreover, the Radon-Nikodym derivative dν ∘ ϕ/dμ is continuous on X′ and, letting S′ = ϕ ⁻¹ Sϕ, we have T _x = S′ ^n(x) x and S′x = T ^m(x) x where n and m are continuous on X′.     

Whitney preserving maps on finite graphs

For a Whitney preserving map f:X→G we show the following: (a) If X is arcwise connected and G is a graph which is not a simple closed curve, then f is a homeomorphism; (b) If X is locally connected and G is a simple closed curve, then X is homeomorphic to either the unit interval [0,1], or the unit circle S¹. As a consequence of these results, we characterize all Whitney preserving maps between finite graphs. We also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.     

Various types of local connectedness in n-fold hyperspaces

Let X be a continuum. The n-fold hyperspace Cn(X), n<∞, is the space of all nonempty compact subsets of X with the Hausdorff metric. Four types of local connectivity at points of Cn(X) are investigated: connected im kleinen, locally connected, arcwise connected im kleinen and locally arcwise connected. Characterizations, as well as necessary or sufficient conditions, are obtained for Cn(X) to have one or another of the local connectivity properties at a given point. Several results involve the property of Kelley or C^*-smoothness. Some new results are obtained for C(X), the space of subcontinua of X. A class of continua X is given for which Cn(X) is connected im kleinen only at subcontinua of X and for which any two such subcontinua must intersect.     

The behavior on fundamental group of a free pro-homotopy equivalence

We study the question: given a morphism $\text{?{(X}{}_{\mathrm{n}}\text{, x}{}_{\mathrm{n}}\text{)}→{(Y}{}_{\mathrm{n}}\text{, y}{}_{\mathrm{<mtext>n</mtext>}}\text{)}}$ in the category pro-(Poi nted. Homotopy) where the domain and range are inverse sequences of well-pointed CW complexes, and given that ? induces an isomorphism {X_n}→{Y_n} in pro-(Homotopy), what additional hypotheses force ? to be an isomorphism in pro-(Pointed Homotopy)? Conjecture. If the dimensions of the Y_n's are bounded, then ? is an isomorphism in pro-(Pointed Homotopy). We first prove the special case of this conjecture in which dim Y_n?d<∞ for all n, and $\text{lim}\text{{H}{}_{\mathrm{d}}\text{Y}{}_{\mathrm{n}}\text{}≠0}$, Y_n being the universal cover of Y_n. Then we deal with the general case: we show that there are certain elements of each π₁Y_n with the properties: (i) these elements commute if and only if ? is an isomorphism in pro-(Pointed Homotopy); (ii) if dim Y_n?d<∞ for all n, then powers of these elements commute. While (i) and (ii) are not incompatible, this result puts severe restrictions on the nature of any possible counter-example to the conjecture.Our method also gives pro-homotopy analogues of the well-known fact that if a K(π, 1) is N-dimensional, then π is torsion-free and contains no abelian subgroup of rank>N. The latter theorems apply to inverse sequences {Y_n} of CW complexes where dim Y_n is finite but not necessarily bounded, hence in particular to infinite-dimensional shape-aspherical compacta.     

Splitting monoidal stable model categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S,S]^C. An idempotent e of this ring will split the homotopy category: [X,Y]^C≅e[X,Y]^C⊕(1−e)[X,Y]^C. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to L_eSC×L_(1−e)SC and [X,Y]^L_eSC≅e[X,Y]^C. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.     

On derived equivalences of categories of sheaves over finite posets

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D^b(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D^b(X) and D^b(Y) are equivalent as triangulated categories.We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence.For any closed subset Y⊆X, we construct a strongly exceptional collection in D^b(X) and use it to show an equivalence D^b(X)?D^b(A) for a finite dimensional algebra A (depending on Y). We give conditions on X and Y under which A becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite .This construction produces many new derived equivalences of posets and generalizes other well-known ones.As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.     

On rectangular products of topological spaces

Several results on rectangular products in the sense of B.A. Pasynkov will be obtained, one of which asserts that for a Tychonoff space X, X × Y is rectangular for any space Y iff X is locally compact and paracompact.     

Metric Möbius geometry and a characterization of spheres

We obtain a Möbius characterization of the n-dimensional spheres S ⁿ endowed with the chordal metric d ₀. We show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to (S ⁿ , d ₀) for some n ≥ 1.     

Lelek maps and n-dimensional maps from compacta to polyhedra

For a natural number m?0, a map from a compactum X to a metric space Y is an m-dimensional Lelek map if the union of all non-trivial continua contained in the fibers of f is of dimension ?m. In [M. Levin, Certain finite-dimensional maps and their application to hyperspaces, Israel J. Math. 105 (1998) 257-262], Levin proved that in the space C(X,I) of all maps of an n-dimensional compactum X to the unit interval I=[0,1], almost all maps are (n−1)-dimensional Lelek maps. Moreover, he showed that in the space C(X,Ik) of all maps of an n-dimensional compactum X to the k-dimensional cube Ik (k?1), almost all maps are (n−k)-dimensional Lelek maps. In this paper, we generalize Levin's result. For any (separable) metric space Y, we define the piecewise embedding dimension ped(Y) of Y and we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a complete metric ANR Y, almost all maps are (n−k)-dimensional Lelek maps, where k=ped(Y). As a corollary, we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a Peano curve Y, almost all maps are (n−1)-dimensional Lelek maps and in the space C(X,M) of all maps of an n-dimensional compactum X to a k-dimensional Menger manifold M, almost all maps are (n−k)-dimensional Lelek maps. It is known that k-dimensional Lelek maps are k-dimensional maps for k?0.